Yes, 4 rotations. I like your explanation.Carbon123 said:The center of mass moves a distance of 2π(r+3r), which is equal to rΘ because it is non slipping,thus Θ=4x2π which is four full rotation.Would this be true ?
The radius of the larger coin in three times that of the smaller coin, not four.TSny said:Yes, 4 rotations. I like your explanation.
Here's a visualization for when the larger coin's radius is 4 times the radius of the smaller coin. https://www.geogebra.org/m/v3a437ux
Yes. The visualization is just another example that I found on the net. It was not meant to represent the setup of the original question. In the visualization, how many revolutions does the small coin make when going once around the larger coin. Is this what you expect?Daniel Gallimore said:The radius of the larger coin in three times that of the smaller coin, not four.
Yes, you are right. The distance traveled by the center of mass is the same as the turning angle multiplied by the radius of the rolling coin.Carbon123 said:Homework Statement
A coin of radius R rolls around a coin of radius 3R which is fixed.If the coin rolls without slipping, How many times does the coin rotates around its axis ?
Homework Equations
Xpm=RΘ for non slipping
The Attempt at a Solution
The center of mass moves a distance of 2π(r+3r), which is equal to rΘ because it is non slipping,thus Θ=4x2π which is four full rotation.Would this be true ? (Some say it is 3 times)
Thanks in advance
Imagine the smaller circle attached at point A on the larger. Cut the larger circle at A to form endpoints A, A'. Keeping A and the attached small circle fixed, unroll the circumference of the larger circle into a straight line AA' of length [itex]6\pi R[/itex]. What you say above then applies: as the smaller circle rolls along the line from A to A' it rotates three times. But now we need to roll the [itex]6\pi R[/itex] line back into being the circumference of a circle, keeping the smaller circle attached at A'. The smaller circle rotates once more in the process.Daniel Gallimore said:The circumference of the larger coin is [itex]2\pi(3R)[/itex] or [itex]6\pi R[/itex]. The circumference of the smaller coin is just [itex]2\pi R[/itex]. If the coin rolls without slipping, then after the coin has rotated once, it should have traversed a distance of [itex]2\pi R[/itex]. Hence, after it has rotated three times, it will have traversed a distance of [itex]6\pi R[/itex], the circumference of the larger coin.
Can you explain why the smaller circle rotates once more in the process of reattaching the endpoints? It seem this operation should have nothing to do with the smaller circle at all. The smaller circle has already rolled across the entire length of the circumference. Many thanks.haruspex said:Imagine the smaller circle attached at point A on the larger. Cut the larger circle at A to form endpoints A, A'. Keeping A and the attached small circle fixed, unroll the circumference of the larger circle into a straight line AA' of length [itex]6\pi R[/itex]. What you say above then applies: as the smaller circle rolls along the line from A to A' it rotates three times. But now we need to roll the [itex]6\pi R[/itex] line back into being the circumference of a circle, keeping the smaller circle attached at A'. The smaller circle rotates once more in the process.
Daniel Gallimore said:Can you explain why the smaller circle rotates once more in the process of reattaching the endpoints? It seem this operation should have nothing to do with the smaller circle at all. The smaller circle has already rolled across the entire length of the circumference. Many thanks.
Yes, I think that's correct if by "displacement" you mean distance traveled. But I hope I'm not overlooking something.Carbon123 said:So it would be correct to say that the displacement of the center of mass is equal to rΘ for any non slipping motion(with respect to a non moving rotating surface )even if the surface is not a line (a curved line,for example) ?
I think I understand. It's essentially the same reason why the moon stays tidally locked with the earth: it's not the the moon isn't rotating; rather, it's rotating just as quickly as it's revolving around the earth. We can hop into the reference frame of the circle as it slides along the surface of the larger coin and perceive no rotation because we are rotating with the circle.TSny said:See post #7 for a nice way to look at it.
Or, suppose you slide the smaller coin around the larger coin so that the smaller coin always has the same point in contact with the edge-surface of the larger coin. Thus, the smaller coin slips around the larger coin without rolling at all. Yet, the smaller coin still makes one rotation while going around the larger coin. You can think of this as the extra rotation that occurs when you include rolling without slipping.
The Rolling Coin Problem is a mathematical puzzle where a coin is placed on a flat surface and a series of moves are made to roll the coin to a specific location. The goal is to determine if it takes 3 or 4 moves to complete this task.
To solve the Rolling Coin Problem, you must first understand the rules of the puzzle. The coin can only be moved by rolling it in a straight line, and it must always touch the surface of the flat surface. Then, using logic and mathematical reasoning, you can determine the minimum number of moves required to roll the coin to the specified location.
Yes, there are various strategies and algorithms that can be used to solve the Rolling Coin Problem. Some common approaches include working backwards from the desired location, breaking the problem into smaller parts, and using mathematical equations to determine the minimum number of moves.
The solution to the Rolling Coin Problem can be determined by using mathematical proofs and logical reasoning. By following a specific strategy and making calculations, you can determine the minimum number of moves required to complete the puzzle. If the minimum number of moves is 3, then the solution is 3 moves. If it is 4, then the solution is 4 moves.
While the Rolling Coin Problem may seem like a purely mathematical puzzle, it does have real-world applications. The problem can be used to model the motion of objects, such as a rolling ball, on a flat surface. It can also be used to develop problem-solving skills and critical thinking in various fields, such as engineering and computer science.